Fundamental Theory

Think outside the box, but connect with reality.

A professor keeps improving his PhD thesis for the rest of his career.

This is a collection of theory projects addressing long-standing questions in physics. I have worked with them on and off for several decades. The following essays and papers are meant to introduce new ideas for solving these problems.

The Stability of the Electron

A longtime favorite of mine has been the question about the internal stability of the electron. If one models an electron as tiny charge cloud, it would explode due to electrostatic repulsion ("Coulomb explosion"). This problem has puzzled top physicists for more than a century, including Poincare, Lorentz, Fermi, Dirac, Feynman, as well as my Diplom thesis advisor Fritz Bopp (the successor of Arnold Sommerfeld at the University of Munich). Analyzing the forces that stabilize the electron might also shed light onto the strength of the electromagnetic force, which is quantified by the electromagnetic coupling constant (the square of the elementary charge). This is the famous fine structure constant alpha, discovered by Sommerfeld in 1916. Feynman once wrote that all good theoretical physicists put the value of alpha up on their wall and worry about it. A balance between an electromagnetic and a non-electromagnetic force would only be possible for a specific value of alpha. That would solve both of these century-old problems in one strike.

Force Balance inside the Hydrogen Atom

Chasing the stability problem led me to many dead ends. Eventually a simple warm-up exercise was successful, where I analyzed the stability of hydrogen atom. The electron is attracted to the positive charge of the proton, threatening to collapse in this case rather than explode. Explaining this puzzle played an important role in establishing quantum mechanics. Today we would say that it is stable, because it represents a minimum of the total energy. The attractive electrostatic energy is balanced by the repulsive kinetic energy generated by squeezing the electron down to the Bohr radius. But such global reasoning does not reveal anything about the local stability of the electron's charge density at a specific point in space.

One can define the local force density acting on the charge cloud of the electron using the Lagrangian formalism (which also serves as the basis of particle theory). It turns out that the electrostatic attraction exactly balances the repulsion from squeezing the electron at every point inside the H atom. At a first glance, such a local force density balance might seem incompatible with the uncertainty relation. But that holds only for a single experiment. If the experiment is repeated many times, one obtains a probability density for the location of the electron -- similar to the force density calculated here. For example, one can determine the probability of finding the electron at a certain location inside the H atom by repeated scattering experiments. Such experiments have been used to determine the charge distribution inside the proton (the so-called form factor).

Stability of a Charged Particle

The H atom contains only a single electron, but there are many situations where one has to deal with the infinite number of "virtual" electrons and positrons that populate the vacuum of quantum electrodynamics (often labeled the Dirac sea). A prototypical example is the vacuum polarization, a faint halo of opposite charges surrounding every charged particle. That creates a situation reminiscent of the H atom, where the electrostatic attraction needs to be balanced by the repulsion from squeezing. But the vacuum polarization is squeezed much harder, down to the reduced Compton wavelength which is a factor of alpha smaller than the Bohr radius (alpha ~ 1/137). Demonstrating the force balance for the infinite number of electrons and positrons in the Dirac sea requires special techniques from quantum electrodynamics to get rid of infinitely large expressions. Since every charged particle is surrounded by the same electron/positron vacuum polarization at large distances, this force is universal.

The figure illustrates the summation of the charge density over all electrons and positrons in the Dirac sea surrounding a positive point charge. These are the spherical wave solutions of the Dirac equation in the Coulomb potential. They are characterized by two quantum numbers, the radial momentum (plotted along the x-axis) and the angular momentum (which increases in steps of 1 from black to magenta). The distance from the point charge is fixed at 1/10 of the reduced Compton wavelength in this plot.

The Stability of the Vacuum Polarization Surrounding a Charged Particle, arXiv:1512.08257 [quant-ph]

The Exchange Force

The problem with Coulomb explosion is solved when going from classical physics to quantum physics. The exclusion principle prevents an electron from interacting with itself. That would imply that the same electron appears twice at the same location, once as the source of its electric field and a second time as the particle that interacts with that field. In Hartree-Fock theory, the resulting self-Coulomb interaction (which causes Coulomb explosion) is canceled by the self-exchange interaction.

Such reasoning brings in the exchange interaction as a major player for determining the stability of the electron. It entangles an electron with the vacuum electrons that form the Dirac sea. Those are able to interact with an extra electron without violating the exclusion principle.

There is an analog to the Dirac sea in solid state physics: the Fermi sea of electrons in a metal. The exchange interaction with the Fermi sea can be explained (and mathematically quantified) by the concept of an exchange hole. It is again the exclusion principle that generates a depletion of electrons in the Fermi sea around an extra electron. The positive charge of the exchange hole compensates the negative charge of the electron and thereby removes the problem with Coulomb explosion.

The analogy with the Fermi sea suggests a similar exchange hole in the Dirac sea. One finds indeed the equivalent of the exchange hole in the Dirac sea. It is about 100-1000 times smaller than that in the Fermi sea and comparable to the reduced Compton wavelength which characterizes the vacuum polarization discussed above. But the exchange hole contains a full unit charge, while the charge induced by vacuum polarization is a factor of alpha smaller.

In addition to the exchange hole the Dirac sea produces an exchange electron. It consists of vacuum electrons displaced by the exchange hole. In the metallic Fermi sea this electron is extracted quickly (think of a metallic wire). But the Dirac sea is an insulator, which forces the displaced electron to stay nearby.

Radial electron densities of the exchange hole, the exchange electron , and
their sum (also shown in 3D as inset). The switch from hole to electron occurs near
the reduced Compton wavelength.

Having figured out the exchange interaction in the Dirac sea, one can now return to the original goal of establishing a force balance for the electron. The electrostatic forces between the exchange hole and its displaced electron will be major players. The magnetic forces are weaker but they grow at small distances.

Anharmonic Waves in (Quantum) Field Theory

This project started with the notion that any realistic theory of the four fundamental interactions has inherently nonlinear character. When fields/particles interact with each other, they violate the superposition principle by generating scattered waves/particles. Those would not exist without nonlinearity. Nevertheless, quantum mechanics is carried out in Hilbert space, a linear vector space describing the various states of a particle. And the current description of the electromagnetic, weak, and strong interactions by quantum field theory operates in Fock space, a collection of linear vector spaces. Gravity, on the other hand, is described by general relativity -- an intrinsically nonlinear theory. That may be the fundamental reason why a theory of quantum gravity in four-dimensional space-time has remained elusive. Would it be possible to formulate our well-tested quantum field theories via a nonlinear version of Fock space that includes gravity?

To flesh out these somewhat vague ideas, I started looking into nonlinear plane waves, first for classical fields and then for quantum fields (see the two links below). Sinusoidal plane waves have served as basis for both classical and quantum fields using Fourier expansions. They are characteristic of linear media, but nonlinear media generate harmonics, a well-known phenomenon in laser physics and acoustics. Therefore my first attempts focused on nonlinear plane waves. Three simple constraints make anharmonic waves compatible with relativistic field theory and quantum physics. But some familiar concepts have to be abandoned, such as the superposition principle, orthogonality, and the existence of single-particle eigenstates.

The figure shows the three possible symmetries of anharmonic waves, with the strength q of the anharmonicity increasing from zero (red) to the maximum (black). The two sets of curves in each plot represent anharmonic generalizations of sine and cosine. In all cases the familiar relation sin2x+cos2x=1 remains valid. This is achieved by a periodic modulation of the phase of sine and cosine. For the lowest symmetry (top row) either the sine or cosine develops a non-zero average which corresponds to a non-zero vacuum expectation value in quantum field theory. This phenomenon enables the Higgs field to give masses to fundamental particles, including itself.

A Higgs Boson Composed of Gauge Bosons

The Higgs boson and the Brout-Englert-Higgs mechanism of symmetry breaking are central to the standard model of physics (therefore a well-deserved Physics Nobel Prize in 2013). However, the Higgs potential of the standard model is an ad-hoc construct that blemishes the elegance of the theory. It contains a quadratic term that mimics a mass term, but with an imaginary mass. A fourth-order term is added, even though no other fundamental particle has such a term. Combined with two adjustable parameters this construct looks rather clumsy. This lack of elegance has been a concern to many theorists and has led to a variety of models where an observable Higgs particle is either absent or composed of fermion pairs (analogous to electron pairs in superconductors). But these models are outperformed by the standard Higgs potential, which matches the data with the help of its adjustable parameters. In all cases one wants to have a finite "vacuum expectation value" (VEV) for the Higgs boson. That is needed to generate masses for fundamental particles. In a fully-symmetric theory they would be massless (like the photon in electromagnetism and the gluons in the theory of the strong interaction).

My contribution to this extensive search is the idea of using pairs of gauge bosons instead of fermion pairs to construct a composite Higgs boson. Gauge bosons are created automatically by gauge symmetry -- an essential part of particle theory. This symmetry eliminates infinities that would otherwise prevent accurate predictions. Each interaction is associated with a characteristic gauge symmetry group, which in turn determines the gauge bosons that mediate the interaction. There is no reason to add a Higgs boson with an adjustable potential. In the standard model the broken symmetry is SU(2). The corresponding gauge bosons are the W and Z which mediate the weak interaction.

At a first glance the concept of gauge bosons with a finite VEV raises concerns, since they are vector particles. If they had a fixed vector as VEV, that would define a preferred direction in space (like a bar magnet pointing in a specific direction). Consequently the rotational symmetry of empty space would be violated. That may be the reason why no one seems to have considered this idea before. But there is an escape: One can choose an individual direction of the VEV for each gauge boson, based on the direction of its momentum (either parallel or perpendicular to it). In quantum field theory, there is an infinite number of virtual gauge bosons floating around in space, each with a different momentum. Since their momenta average out to zero, their VEVs do so, too. That eliminates any preferred direction in space. Nevertheless, there will be a preferred direction in the neighborhood of a specific gauge boson -- more precisely, within its Compton wavelength h/Mc (h = Planck's constant, M = mass of the gauge boson, c = velocity of light). Since gauge symmetry is a local symmetry, it can be broken locally by the VEV of an individual gauge boson. The interactions of a gauge boson with other particles are local as well -- and in particular its self-interactions. Those determine the VEV in the proposed model.

The connection between the scalar VEV of the standard Higgs boson and the vector VEVs of the gauge bosons is established by comparing their squares (which are scalars). This concept immediately produces a formula for the mass of the Higgs boson. It is simply half of its VEV. The latter can be obtained directly from the experimental value of Fermi's coupling constant GF. The result matches the observed Higgs mass within 2% , the accuracy expected for the "tree-level" approximation used in this work. The standard model, as well as other Higgs models are not able to explain the Higgs mass. That's why it took so long to find the Higgs boson within the wide mass range allowed by theoretical constraints. The new model also allows the parameters of the Higgs potential to be calculated from the fundamental self-interactions of the gauge bosons, thereby providing additional tests. However, that would require an expert in writing the sophisticated computer programs that are required to calculate quadratic and quartic self interactions in a Higgs-free gauge theory.

As it stands, this model can serve as testbed for replacing the artificial Higgs potential by fundamental gauge interactions. Before using it as alternative to the standard model, one will have to incorporate fermions into the model. To get started, one could use the Higgs-fermion interaction of the standard model, but there might be more elegant options that avoid introducing all the fermion masses as adjustable parameters. As long as the Higgs-fermion interaction remains an open question, one can try to construct an empirical Higgs potential from observables, such as the masses of the gauge bosons and Fermi's coupling constant. As shown in the publication below, this is indeed possible. Since the Higgs boson consists of the W and Z gauge bosons, the Higgs potential can be plotted as a function of the the scalar products WW and ZZ, as shown in the figure. There are two topologically distinct cases, with the minimum occuring either along a line or at a point.

The two options for the potential of the composite Higgs boson, plotted in terms of the W and Z gauge boson potentials (in units of GeV). Such potentials replace the traditional Higgs potential. They can be calculated from the gauge boson self-interactions.